Balkan Mathematical Olympiad (BMO)

The sum of the first n positive integers is given by the formula n(n+1)/2. Therefore, the sum of the first 100 positive integers is 100(101)/2 = 5050.

The area of a circle is given by the formula πr^2. Therefore, the area of a circle with radius 5 cm is π(5)^2 = 25π cm^2.

The volume of a cube is given by the formula s^3, where s is the side length. Therefore, the volume of a cube with side length 3 cm is (3)^3 = 27 cm^3.

The equation of a line that passes through two points (x1, y1) and (x2, y2) is given by the formula y - y1 = (y2 - y1)/(x2 - x1) * (x - x1). Substituting the values of the two points, we get y - 3 = (7 - 3)/(5 - 2) * (x - 2). Simplifying this equation, we get y = 2x + 1.

We can solve this equation using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. Substituting the values of a, b, and c, we get x = (-(-4) ± √((-4)^2 - 4(1)(3))) / 2(1). Simplifying this equation, we get x = 1, x = 3.

When flipping a coin, there are two possible outcomes: head or tail. Since these outcomes are equally likely, the probability of getting a head is 1/2.

The derivative of a function f(x) is given by the formula f'(x) = lim_(h->0) [f(x+h) - f(x)] / h. Substituting the values of f(x), we get f'(x) = lim_(h->0) [(x+h)^3 - 2(x+h)^2 + 3(x+h) - 4 - (x^3 - 2x^2 + 3x - 4)] / h. Simplifying this equation, we get f'(x) = 3x^2 - 4x + 3.

The integral of a function f(x) from a to b is given by the formula ∫[f(x) dx] from a to b. Substituting the values of f(x), a, and b, we get ∫[2x + 3 dx] from 0 to 2. Simplifying this equation, we get 14.

The expression log_2(16) means the exponent to which 2 must be raised to get 16. Since 2^4 = 16, the value of the expression is 4.

The expression sin(π/2) means the sine of the angle π/2. Since the sine of an angle is the ratio of the opposite side to the hypotenuse of a right triangle, and the opposite side of a right triangle with an angle of π/2 is equal to the hypotenuse, the value of the expression is 1.

The expression cos(π) means the cosine of the angle π. Since the cosine of an angle is the ratio of the adjacent side to the hypotenuse of a right triangle, and the adjacent side of a right triangle with an angle of π is equal to the opposite side, the value of the expression is -1.

The expression tan(π/4) means the tangent of the angle π/4. Since the tangent of an angle is the ratio of the opposite side to the adjacent side of a right triangle, and the opposite side and the adjacent side of a right triangle with an angle of π/4 are equal, the value of the expression is 1.

The expression sec(π/3) means the secant of the angle π/3. Since the secant of an angle is the ratio of the hypotenuse to the adjacent side of a right triangle, and the hypotenuse and the adjacent side of a right triangle with an angle of π/3 are equal, the value of the expression is 2.

The expression csc(π/6) means the cosecant of the angle π/6. Since the cosecant of an angle is the ratio of the hypotenuse to the opposite side of a right triangle, and the hypotenuse and the opposite side of a right triangle with an angle of π/6 are equal, the value of the expression is 2√3.

The expression cot(π/4) means the cotangent of the angle π/4. Since the cotangent of an angle is the ratio of the adjacent side to the opposite side of a right triangle, and the adjacent side and the opposite side of a right triangle with an angle of π/4 are equal, the value of the expression is 1.